When vandals defaced a Confederate memorial at a Los Angeles cemetery last week, retiree Scarlett Stahl took it personally, as if her own property had been damaged.

In a way, it had. She heads the California division of the United Daughters of the Confederacy, a national organization whose members donated the large stone and bronze monument to Hollywood Forever Cemetery in 1925 and was the driving force behind scores of the Confederate monuments across the U.S. that are now being decried as symbols of bigotry and are being...